&nbspReflection: Backwards Planning The Depths and Death of Space - Section 3: Is he correct?

My students are not familiar with scientific notation yet, at least we have not covered it to date in my course. Today, they will encounter it as they use calculators to deal with this problem. I view this as a great opportunity to discuss scientific notation and its appearance on the calculator. I am ready and I will be on the lookout for students staring at a calculator with a puzzled face.

I cover Scientific Notation later in my course in the following units:

I read them the article from Wired.com and discuss the meaning of a dying star. I focus them on the last paragraph of the article:

"The observations amounted to nearly eight days of exposure time over a 67-day period. This allowed for even fainter dwarfs to become visible, until at last the coolest — and oldest — dwarfs were seen. These stars are so feeble...they are less than one-billionth the apparent brightness of the faintest stars that can be seen by the naked eye (WIRED.com, accessed September 26 2013)."

I ask them to write down "one-billionth" in their notebook and take 2 minutes to describe what that means. They can use any method, but I want to see their interpretation. Students will write stuff like, "imaging you took something and broke it into a thousand thousand thousand pieces." Some will go directly to the numbers in decimal or fraction form, but the goal is to have a quick share about this immense number. "I like to extend it and ask, what about a trillionth?" We also make quick mention of the "th" at the end of the word, connecting billion and billionth, trillion and trillionth and so forth.

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Isaac Asimov's short story, "The Last Question," explore the future of humanity starting in the late 21st century. In this not to distant future, a computer runs all day to day operations on Earth and manages to solve our energy crisis by utilizing solar rays and converting them to whatever energy needs we have.

I have a copy of Asimov's Robot Dreams, which contains the story. I like to have the students read it for about 15 minutes and then I bring up a few questions, but I always consult with the humanities department first (as any math teacher should). I need to get their advice on having the group read and ask them to read it as well. Sometimes we agree that it is better for me to read it or summarize it. Other times I have students read the story before they come to class. It all depends on the group I have at the time and the strategies that the humanities teachers are currently using. Sometimes I am lucky and able to incorporate the short story into their fiction reading units.

Basically I start by describing the premise as I did above, and then continue into the heart of the story:

"When multivac, the super computer solved the energy crisis, one worker exclaims, 'now our energy crisis is solved forever!' But his friend says, 'not forever.' The idea here is that no matter how long we are able to get energy from the sun, it will run out sometime (approximately 5.4 billion years)."

Again I ask them to pause and write this number out, "What do you think of when you hear 5.4 billion?"

Then I resume the story, and the plot continues to where many humans at different points in our future contemplate how long we can last in this universe. At one key point in the story, two humans living over 20,000 years from now are asking the same question, "how long will we last?" Their names are VJ-23X and MQ - 17J.

"Space is infinite, A hundred billion galaxies are there for the taking. More."

"A hundred billion is not infinite and its getting less infinite all the time. Consider! Twenty thousand years ago, mankind first solved the problem of utilizing stellar energy, and a few centuries later, interstellar travel became possible. It took mankind a million years to fill one small world and then only fifteen thousand years to fill the rest of the Galaxy. Now the population doubles every ten years-"

VJ-23X interrupted, "We can thank immortality for that."

"Very well. Immortality exists and we have to take it into account."

"But to get back to my point. Population doubles every ten years. Once this galaxy is filled, we'll have filled another in ten years. Another ten years and we'll have filled two more. Another decade, four more. In a hundred years, we'll have filled a thousand galaxies. In a thousand years, a million galaxies? In ten thousand years, the entire known Universe. Then what?"

Here the question becomes, "is this correct? How long would it take to fill all the galaxies in the universe? And what would the population be?"

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The problem is, "if human population is doubling every ten years, how long will it take to fill all the galaxies in the universe and how many people will there be?" To get the class started with the questions that they need to answer here, I ask them "what do you need to know in order to solve this problem? Are there any assumptions that you will need to make?"

Here are some assumptions that help us:

1. At this point in the story, humans are immortal.

2. There are about 100 billion galaxies

3. There are between 100-400 billion planets in our galaxy

4. There are almost 7 billion people on Earth today

As students begin to explore the question above, I continue to ask questions about their assumptions and strategies. The answer is important, but I want them to understand that every assumption can have a dramatic change on the answer. For example, I might ask:

Are there 100 billion planets in every galaxy?

Would every planet hold 7 billion people?

Do you agree that there are only 100 billion galaxies?

At this point the students are thinking hard, getting a good workout with really big numbers.

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At some point, it is really important to bring the class back together and discuss the work of using math to model this situation and try to answer the given question. In doing so, my goal is to compare answers and discuss assumptions made. I want the whole class to think about how the assumptions made within each group made a major impact on the group's answer.

We finish the discussion by comparing our values to the notion of infinity. The questions here are fundamentally human:

Can we expand our population forever?

How long can we grow our population?

Is there an end?

Will the galaxies last long enough for us to reach them (assuming this is possible)?